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Philosophy of Science

The Relational Nature of Species Concepts

José E. Burgos
Universidad Central de Venezuela,
Universidad Católica Andrés Bello

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ABSTRACT: Édouard Le Roy as early as 1901 observed the existence of an intellectual movement seeking to break from traditional positivism and set for himself the task of drawing up the program of this new positivism. Noting that this program precedes the Vienna Circle, I endeavor to determine its nature and to evaluate its impact on logical positivism. Viewed in this light, the discussions between Le Roy, Poincaré and Duhem appear more prolonged and substantial than is usually thought. What we have here is perhaps not a homogeneous doctrine but a vigorous intellectual movement, from which logical positivists were able to borrow specific theses in their attempts to mitigate Mach's strict positivism; more important still, they had before them an example of neopositivism. History is not the only concern: among the issues debated, one encounters the claim that facts are theory-laden. This claim still stirs controversy today. An inquiry into the origins of the claim is one way of clarifying the arguments involved.

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The definition of the concept of a species has been a recurrent issue in evolutionary biology at least since the publication of The origin of species (Darwin, 1859). Despite numerous treatments (e.g., Ghiselin, 1974, 1987; Hull, 1976a, 1976b, 1978; Kitcher, 1984; Kitts, 1984; Kitts & Kitts, 1979; Mayr, 1957, 1963, 1976a, 1976b, 1987; Meglitsch, 1954; Mishler & Brandon, 1987; Mishler & Donoghue, 1982; Sober, 1984; Sokal, 1973; Sokal & Crovello, 1970; Wiley, 1978), the species problem still represents a conceptual issue.

In the present paper, I submit that all species concepts are relational in nature. In its most general expression, this idea is not new. Indeed, according to Mayr (1976a), "the species concept is a relational concept" (p. 480). However, the interpretation and further elaboration I propose is different from Mayr's. The crucial difference is that I adopt the set-theoretic notion of a relation. A consequence of this adoption is that, in contrast to Mayr, who restricts his idea to the relation of reproductive isolation (which defines his own species concept), I argue that the set-theoretic notion of a relation allows us to define any species concept in terms of some kind of relation. A second consequence has to do with the rejection expressed by Mayr and others towards viewing species as sets. I shall have something to say about this issue at the end of the paper.

The notion of a relation is central to scientific concepts in general, and the species concept is no exception. But what is a relation? We all have an intuitive idea. However, our intuitive ideas are useful up to a point. If one proposes a philosophical thesis according to which the notion of a relation is central to all species concepts, one should use the most precise definition available. Set theory provides us with such a definition, for which I shall adopt it here. As I will show, this notion allows for a rather unequivocal definition and further comparison of different species concepts, within a common framework.

Relational Sentences

The starting point towards showing the suitability of the set-theoretic notion of a relation to elucidate the nature of species concepts is to acknowledge the existence of relational sentences and their ubiquity in scientific language. A relational sentence specifies at least two proper names separated by a term that expresses some kind of a connection, association, or link between them. Proper names identify or represent individuals, that is, discrete entities that retain a certain identity across spacetime, an identity that, in principle, can be specified to some degree of accuracy. What analytic philosophers call "individuals" is closely exemplified by what biologists call "specimens". Typically, the term "specimen" is used to refer to a concrete, particular, specific organism that has been obtained, picked, or chosen through a relatively well-defined procedure, and is kept under some kind of observation. Specimens can be dead or alive, and, when alive, they can be in captivity or roaming freely about their natural environments.

If specimens are identified through some labeling system, then we can describe them in certain ways. Take, for example, the following sentences:

1. "a is a male"
2. "b has two legs"
3. "c has a 1-cm long beak"
4. "d has feathers"

where a, b, c, and d are labels that identify certain specimens. Such sentences certainly play a crucial descriptive role in evolutionary biology. However, they do not arise from, exemplify, or presuppose any species concept. To be sure, Sentence 1 presupposes the male-female dichotomy, which is central to the biological species concept. However, such a dichotomy does not define this concept. That is, according to this concept, Specimen a is not a member of a particular species just because it is a male. Similarly, according to the phylogenetic or phylogenetic concept, d is not a member of a certain species just because it has feathers. The same applies to other descriptive sentences and species concepts. Instead, species concepts arise from relational sentences, which express connections between specimens. Compare the following sentences with the previous ones:

5. "a breeds with b".
6. "a is phenotypically similar to b, to a degree n".
7. "a descends from c".
8. "c is an ancestor of a".

I contend that these kinds of sentences are central to an analysis of any species concept.

The order in which proper names occur throughout a relational sentence can make an important difference. For example, under the biological concept, the sentence "a breeds with b" seems to imply the sentence "b breeds with a", for which the order of a and b does not seem to be particularly relevant. However, under the phylogenetic concept, Sentence 7 does not imply "c descends from a", but Sentence 8. Note that the only difference in these two pairs of sentences is the order of appearance of a and b. This order can be expressed with the set-theoretic notion of a relation in terms of ordered pairs.

Ordered Pairs

An ordered pair is a set of the form (x,y), where x and y represent whatever individuals are being considered at a particular moment. The ordered pair corresponding to Sentences 5 and 6, then, would be (a,b), whereas the order pair corresponding to the sentence "b breeds with a" would be (b,a). Set theory stipulates that (a,b) is different from (b,a), although under the biological concept such a difference is irrelevant. Also, the ordered pair corresponding to Sentence 7 would be (a,c), whereas the ordered pair corresponding to the sentence "c descends from a" would be (c,a). Again, within set theory, (a,c) is different from (c,a), and (in contrast to the biological concept) such a difference is crucial under the phylogenetic concept. In general, by stipulating that (x,y) is different from (y,x), set theory allows us to distinguish between relational sentences in terms of the order of appearance of their constituting proper names, a distinction that can be relevant under certain species concepts.

The notion of an ordered pair also allows for the repetition of proper names. For example, the sentence "a breeds with a" is syntactically correct, although it may be semantically incorrect under the biological concept, insofar as this concept forbids the possibility of a specimen breeding with itself. However, the sentence "a is phenotypically similar to a" makes perfect sense (although a trivial one) under the phenetic concept. The ordered pair corresponding to such sentences would be (a,a).


According to the set-theoretic notion, a relation is set of ordered pairs. Viewed as a set, a relation can be defined in two ways, namely, by extension or by abstraction. A relation is defined by extension when a list of all of its constituting ordered pairs is specified. For example, the expression

R = {(a,b), (c,d), (a,f)}

stipulates in set-theoretic notation that R identifies the relation whose elements are (a,b), (a,c), and (a,d). Why should we consider those pairs as constituting a set? The answer will depend upon two factors: First, the meaning of the proper names (exactly what kind of individuals do they label); and, second, the relational sentences in which they appear. Regarding the first factor, I have already said that, in the present context, proper names identify or label specimens. The second factor requires more elaboration.

Take again Sentences 5 to 8. Although they refer to exactly the same specimens, they do not have much in common, for their respective relational terms (viz., "breeds with", "is phenotypically similar to", "descends from") mean different things. So much, that they determine different species concepts. This idea of a determination of different species concepts by different relational terms is central to my proposition. I propose, then, that the distinction between the biological and phylogenetic concepts corresponds very closely to the distinction between the meanings of the relational terms "breeds with" and "descends from", which, in turn, determine different kinds of relations. Also, the distinction between the pheneticist concept and the phylogenetic concept corresponds very closely to the distinction between the meanings of the relational terms "is phenotypically similar to" and "descends from", and so on. It is the meaning of a relational term which determines a particular species concept. And this is common to all species concepts. Hence the relational nature of species concepts. Let me illustrate this idea with the biological concept.

Application to the Biological Concept

The biological concept arises from the relational term "is reproductively isolated from", which is a compound of the more basic terms "breeds with" and "is geographically isolated from". First, I will show how we can define relations through "breeds with". Then, I will show how (the negation of) "is geographically isolated from" must be introduced to make classification possible. I shall adopt the following definition of "breeds with":

"A specimen x breeds with another specimen y if and only if:

x is either a male or a female,
if x is a male, then y is a female,
if x is a female, then y is a male;
x and y copulate;
x and y produce viable offspring".

My aim here is not to defend this particular definition of the term "breeds with", but only to show how the set-theoretic notion of a relation is sufficient to define the biological concept, given some definition of certain relational terms.

The above definition provides us with a criterion to decide whether or not any given pair of specimens (x,y) qualifies as a member of a breeding relation. Such membership depends upon whether or not such a pair transforms the schema "x breeds with y" into a true relational sentence. For example, if x = a and y = b, then the schema is transformed into "a breeds with b". If this sentence is empirically true, then the pair (a,b) is a member of a breeding relation. A breeding relation then is a set of ordered pairs of specimens that transform the schema "x breeds with y" into a true sentence.

For example, let a, b, c, d, e, and f be a given sample of specimens, such that a, c, and e are males, and b, d, and f are females. Suppose also that the following sentences are true, according to our criterion:

"a breeds with b",
"c breeds with d",
"e breeds with f".

Pairs (a,b), (c,d), and (e,f), then, constitute a breeding relation. We can thus write

B = {(a,b), (c,d), (e,f)}

to define that relation by (an observed) extension.

We can construct many other sentences and ask if they are true according to our criterion. Take the following:

"b breeds with a",
"d breeds with c",
"f breeds with e".

Are (b,a), (d,c), and (f,e) also members of B? From our definition of "breeds with", it seems that this question would be answered affirmatively, for if a breeds with b, then b breeds with a, and so on Therefore, those three pairs also qualify as members of B, for which we can rewrite:

B = {(a,b), (c,d), (e,f), (b,a), (d,c), (f,e)}.

This definition may seem redundant, for we are talking about the very same specimens in (a,b) and (b,a). However, if we focus upon the pairs rather than the individual specimens, that redundancy expresses an important property of B, namely symmetry. Another property of relations is reflexivity, which involves asking whether or not the following sentences are true:

"a breeds with a",
"b breeds with b", ...

According to our definition, no specimen can breed with itself, for which these sentences would be false and (a,a), (b,b), et cetera do not qualify as members of B. Hence, B lacks a property of relations, namely reflexivity. In general, then, breeding relations are nonreflexive and symmetric.

A third property of relations is transitivity. To determine whether or not B is transitive, we need to consider three different specimens at a time. For example, suppose that a also breeds with f, for which (a,f) and (f,a) would also be members of B and we can rewrite:

B = {(a,b), (c,d), (e,f), (b,a), (d,c), (f,e), (a,f), (f,a)}.

Now, if f breeds with a and a breeds with b, does this imply that f breeds with b? In other words, is (f,b) a member of B just by virtue of (f,a) and (a,b) being members of B? We can answer this question negatively on purely logical grounds, for b and f are females. Of course, we can always test our inference empirically by observing whether or not b and f actually breed. If they do, then our definition of "breeds with" must be revised. But if they do not, then B is nontransitive.

On the basis of the above analysis, we can say that a breeding relation represents a species, and that the biological species category is the set of all breeding relations. However, the application of our membership criterion to any sample of specimens will result in only two breeding relations, namely the set of pairs that transform "x breeds with y" into a true sentence, and the set of pairs that transform this schema into a false sentence. Indeed, if a and b are blue jays, c and d are wolves, and e and f are humpback whales, then B would be a single breeding relation consisting of a pair of blue jays, a pair of wolves, and a pair of humpback whales. And if we add a pair of breeding Siamese Fighters g and h to our specimen sample, then (g,h) would also be a member of B. All of these pairs would thus represent a single species, which is counterintuitive from a systematist's perspective. How can we define several breeding relations, so that classification under the biological concept is possible?

The answer is to add a relational term to our membership criterion. Under the biological concept, the relational term of choice would be "is geographically isolated from". However, this term results in sets that do not represent species. So, we need to introduce the term "is not geographically isolated from" or "inhabits the same geographic area as" to obtain breeding relations that do represent species. We can thus stipulate that two pairs of specimens (w,x) and (y,z) are members of the same breeding relation if and only if:

w breeds with x,
y breeds with z,
w breeds with z,
y breeds with x,

(w,x) inhabits the same geographic area as (y,z).

Note that "inhabits the same geographic area as" relates pairs of specimens, rather than individual specimens. If we apply this term to individual specimens we will produce the same problem we are trying to solve.

If we rely upon an intuitive idea of what inhabiting the same geographic area is, (a,b), (c,d), (e,f), and (g,h) would not constitute a single breeding relation anymore, insofar as these pairs would be expected to be geographically isolated from one another, at least by their natural habitats. As a result, we have four breeding relations representing four species:

B1 = {(a,b), (b,a)},
B2 = {(c,d), (d,c)},
B3 = {(e,f), (f,e)},
B4 = {(g,h), (h,g)}.

We can add more specimens to our sample. Suppose we add i and j, so that "i breeds with j" is true. If (i,j) inhabits the same geographic area as (a,b), then (i,j) and (j,i) are members of B1. If (i,j) is geographically isolated from the other specimen pairs, then (i,j) and (j,i) would constitute a separate breeding relation (viz., B5), and so on.

Concluding remarks

I have proposed the set-theoretic notion of a relation as a tool for clarifying the nature of species concepts. Under this notion, different species concepts become unitary regarding their relational nature, but pluralistic regarding the kinds of relations that define them. Also, the notion leads to the idea that members of species are pairs of specimens, rather than individual specimens. Finally, the example application suggests that the set-theoretic notion is sufficient to capture many aspects of the biological concept. I believe the same applies to other concepts. Differences and similarities among species concepts thus arise naturally. For example, in contrast to reproductive isolation, and relying on an intuitive meaning of the terms "is phenotypically similar to" and "descends from", phenotypic similarity seems to be reflexive, symmetric, and transitive, whereas the relation of descent seems to be only transitive. These kinds of comparisons open new possibilities for conceptual and empirical research.

I would like to end by commenting briefly on the rejection shown by some authors towards applications of set theory to the species concept (e.g., Hull, 1978; Mayr, 1987). The basic argument behind such a rejection is roughly the following. Sets are inherently static, whereas species are inherently dynamic. Hence, species are not sets. The main difficulty with this argument is that the static-dynamic distinction is inextricably linked to the concept of time. But this concept is not part of set theory, for which sets are not inherently static (or dynamic).

The set-theoretic notion of a relation, however, allows us to talk about dynamic phenomena if we use specific temporal locations as defining properties of relations. Indeed, we can define a phylogenetic history as a sequence (i.e., an ordered set) of sets of organisms, each one defined, at least in part, by a particular time interval. If we use time as a defining property of particular sets, we can also link the biological and the phylogenetic concepts by defining sequences of breeding relations across time that are connected through a descent relation. We can also use specific spatial locations as defining properties of relations. Set theory thus allows for the definition of sets that represent spatiotemporally restricted aggregates of organisms.

In this manner, the set-theoretic notion of a relation allows us, at least in principle, to talk about different species concepts, thus providing us with a common linguistic framework to define, discuss, compare, and evaluate different species concept. The notion has the advantages of simplicity, clarity, and expressive power, which facilitate communication and conceptual unification.

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